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What is the period of a state in a Markov chain?
Periodicity (source: wikipedia) A state i has period k if any return to state i must occur in multiples of k time steps. Formally, the period of a state is defined as. k = gcd{n:Pr(Xn=i|X0=i)>0} (where “gcd” is the greatest common divisor).
What is the period of a state?
The period of a state i is the largest integer d satisfying the following property: p(n)ii=0, whenever n is not divisible by d. The period of i is shown by d(i). If p(n)ii=0, for all n>0, then we let d(i)=∞.
Do transient states have periods?
In a class, all the states have the same period. In some article, by definition A has a period=0. It’s a transient state. B and C have a period of one ( there is loop on themselves).
What are classes in Markov chain?
in which each subset has the property that all states within it communicate. Each such subset is called a communication class of the Markov chain. If we now consider the rat in the closed maze, S = {1,2,3,4}, then we see that there is only one communication class C = {1,2,3,4} = S: all states communicate.
When does a Markov chain have period 1?
A Markov Chain is aperiodic if all states have period 1. In your example, it’s possible to start at 0 and return to 0 in 2 or 3 steps, therefore 0 has period 1.
How many states are in a Markov chain?
Now imagine that the clock represents a markov chain and every hour mark a state, so we got 12 states. Every state is visted by the hour hand every 12 hours (states) with probability=1, so the greatest common divisor is also 12. So every (hour-)state is periodic with period 12.
Why is the period of a stochastic process aperiodic?
Here is the problem from a stochastic processes book: The answer in the book states that the period is 1, and therefore the chain is aperiodic. But I do not understand why the period is not 2.
Which is the most common Markov chain in scalar multiplication?
10 MARKOV CHAINS: BASIC THEORY. scalar multiplication by positive integers, that is, if m 2Ax then all multiples of m are elements of Ax . Second, the greatest common divisor of the numbers in Ax is 1, because by hypothesis the Markov chain is aperiodic.